Initial-state-independent equilibration at the breakdown of the eigenstate thermalization hypothesis

Khodja, Abdellah; Schmidtke, Daniel; Gemmer, Jochen

doi:10.1103/physreve.93.042101

Cited by 4 publications

(35 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Other investigations address the question if physical quantities like expectation values dynamically approach constant equilibrium values that are consistent with corresponding ensemble values [6,7]. The works [8][9][10][11][12] indicate that equilibrium values can depend on the concrete initial state under certain conditions. One approach to this topic is the eigenstate thermalization hypothesis (ETH) [13][14][15].…”

mentioning

confidence: 99%

“…Furthermore, we set J = 1 and ∆ = 0.1 throughout the following investigations. We focus on two cases of weak (J c = 0.2) and strong (J c = 4.5) inter-chain coupling, which represent two particular regimes for this model [10,11]. To keep the total energy approximately constant we fix β = 1 and analyze the initial state dependence by varying δ, such that the regarded initial expectation values cover the whole spectrum of A.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Necessity of eigenstate thermalisation for equilibration towards unique expectation values when starting from generic initial states

Bartsch¹,

Gemmer²

2017

EPL

View full text Add to dashboard Cite

We investigate dynamical equilibration of expectation values in closed quantum systems for realistic non-equilibrium initial states. Thereby we find that the corresponding long time expectation values depend on the initial expectation values if eigenstate thermalization is violated. An analytical expression for the deviation from the expected ensemble value is derived for small displacements from equilibrium. Additional numerics for magnetization and energy equilibration in an asymmetric anisotropic spin-1/2-ladder demonstrate that the analytical predictions persist beyond the limits of the theory. The results suggest eigenstate thermalization as physically necessary condition for initial state independent equilibration. One approach to this topic is the eigenstate thermalization hypothesis (ETH) [13][14][15]. The ETH implies that expectation values of an observable of interest A with regard to energy eigenstates |n vary slowly as a function of the corresponding eigenenergies E n , which means that all diagonal elements of A in the energy eigenbasis, i.e., A nn = n|A|n , are approximately constant within some energy regime which the system's state is restricted to. For open system situations, i.e., when the global system is divided into system plus bath, there is an alternative formulation of the ETH. If the reduced density matrices of eigenstates of the global system Tr B {|n n|} vary slowly as a function of the global eigenenergies E n in terms of a suitable operator norm, then the ETH is valid. This is equivalent to the statement that the previous definition is fulfilled for all observables acting solely on the systemIf the ETH is fulfilled, then there is initial state independent (ISI) equilibration of expectation values with regard to A [15]. If the ETH is not fulfilled, there are still initial states which yield ISI equilibration. They even form the majority of possible initial states in a statistical sense according to the Haar measure [6,16]. However, in those cases it is usually required that the Hamiltonian, the regarded observable and the initial state are uncorrelated meaning that the mutual orientations of the eigenbases of those operators can be viewed as more or less random and therefore statistically independent [6,17].While this assumption is mathematically reasonable, in the sense of a high relative frequency with respect to the unitary Haar measure, its physical justification is at the heart of the present investigation. Moreover, some investigations rely on rather strong restrictions on the initial state like identical amplitudes on all energy eigenstates in the regarded energy shell [17,18].In this Letter we address the question whether the validity of the ETH is needed for ISI equilibration of expectation values in a realistic setting, i.e., under realistic modeling of the initial state. To this end we first introduce a parametrized class of initial states (1) and derive an analytical expression which quantifies the initial state dependence for small deviations from equilibrium. Furthermore...

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

Necessity of eigenstate thermalisation for equilibration towards unique expectation values when starting from generic initial states

Bartsch¹,

Gemmer²

2017

EPL

View full text Add to dashboard Cite

show abstract

“…7 (c) presents a finite-size scaling of R M = (M ∞ − M eq )/(M * − M eq ) for different values of the external force α. In particular, we also show data for the chaotic region of the parameter space with smaller rung couping J ⊥ = 0.2, where the ETH is expected to apply [31,44,45]. [Note however, that the LRT regime is found to be smaller in the chaotic regime, see inset of Fig.…”

Section: The Asymmetric Spin Laddermentioning

confidence: 92%

“…Based on a finite-size analysis of level statistics and of fluctuations of diagonal matrix elements, the spin ladder (16) has been shown to undergo a transition between a chaotic phase and an ETH-violating phase for large interchain couplings J ⊥ /J 4 [44,45]. Since our goal is not to thoroughly screen all parameter regimes, but rather to numerically illustrate the physical mechanisms discussed in Sec.…”

Section: The Asymmetric Spin Laddermentioning

confidence: 99%

Relaxation of dynamically prepared out-of-equilibrium initial states within and beyond linear response theory

et al. 2019

Self Cite

View full text Add to dashboard Cite

We consider a realistic nonequilibrium protocol, where a quantum system in thermal equilibrium is suddenly subjected to an external force. Due to this force, the system is driven out of equilibrium and the expectation values of certain observables acquire a dependence on time. Eventually, upon switching off the external force, the system unitarily evolves under its own Hamiltonian and, as a consequence, the expectation values of observables equilibrate towards specific constant long-time values. Summarizing our main results, we show that, in systems which violate the eigenstate thermalization hypothesis (ETH), this long-time value exhibits an intriguing dependence on the strength of the external force. Specifically, for weak external forces, i.e., within the linear response regime, we show that expectation values thermalize to their original equilibrium values, despite the ETH being violated. In contrast, for stronger perturbations beyond linear response, the quantum system relaxes to some nonthermal value which depends on the previous nonequilibrium protocol. While we present theoretical arguments which underpin these results, we also numerically demonstrate our findings by studying the real-time dynamics of two low-dimensional quantum spin models.

show abstract

“…Nonintegrable spin ladders (in the sense of the Bethe ansatz) and derivates of it are already intensively studied regarding e.g. relaxation of magnetization [17,18,20] or energy imbalances [42,43] and thus are ideal systems to start with. Note that, in order to allow for a non-trivial resonant driving protocol, we apply also a static magnetic field on the additional spin (see for details below).…”

Section: Models Initial States and Driving Protocolmentioning

confidence: 99%

Stiffness of probability distributions of work and Jarzynski relation for non-Gibbsian initial states

Schmidtke¹,

Knipschild²,

Campisi³

et al. 2018

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

We consider closed quantum systems (into which baths may be integrated) that are driven, i.e., subject to time-dependent Hamiltonians. Our point of departure is the assumption that if systems start in non-Gibbsian states at some initial energies, the resulting probability distributions of work may be largely independent of the specific initial energies. It is demonstrated that this assumption has some far-reaching consequences, e.g., it implies the validity of the Jarzynski relation for a large class of non-Gibbsian initial states. By performing numerical analysis on integrable and nonintegrable spin systems, we find the above assumption fulfilled for all examples considered. Through an analysis based on Fermi's golden rule, we partially relate these findings to the applicability of the eigenstate thermalization ansatz to the respective driving operators.

show abstract

Initial-state-independent equilibration at the breakdown of the eigenstate thermalization hypothesis

Cited by 4 publications

References 41 publications

Necessity of eigenstate thermalisation for equilibration towards unique expectation values when starting from generic initial states

Necessity of eigenstate thermalisation for equilibration towards unique expectation values when starting from generic initial states

Relaxation of dynamically prepared out-of-equilibrium initial states within and beyond linear response theory

Stiffness of probability distributions of work and Jarzynski relation for non-Gibbsian initial states

Contact Info

Product

Resources

About